Question

use the empirical rule to answer the following questions.
a. the mean is.
b. the standard deviation is.
c.% of the test scores are between 56 and 66.
d.% of the test scores are between 51 and 71.
e.% of the test scores are between 46 and 76.
f.% of the test scores are between 61 and 66.
g.% of the test scores are less than 61.
h.% of the test scores are less than 66.

Explanation:

Step1: Find the mean

The mean of a normal distribution is at the center of the curve. From the graph, the center is at \(61\).

Step2: Find the standard deviation

The distance between the mean and the first mark on either side is the standard deviation. \(61 - 56=5\) or \(66 - 61 = 5\), so the standard deviation \(\sigma=5\).

Step3: Use the Empirical Rule for part c

The Empirical Rule states that approximately \(68\%\) of the data is within \(1\sigma\) of the mean (\(\mu\pm\sigma\)), \(95\%\) within \(2\sigma\) (\(\mu\pm2\sigma\)), and \(99.7\%\) within \(3\sigma\) (\(\mu\pm3\sigma\)). For \(56\) to \(66\) (\(\mu - \sigma\) to \(\mu+\sigma\)), it's \(68\%\).

Step4: Use the Empirical Rule for part d

For \(51\) to \(71\) (\(\mu - 2\sigma\) to \(\mu + 2\sigma\)), it's \(95\%\).

Step5: Use the Empirical Rule for part e

For \(46\) to \(76\) (\(\mu-3\sigma\) to \(\mu + 3\sigma\)), it's \(99.7\%\).

Step6: Use the Empirical Rule for part f

Since \(61\) to \(66\) is half of \(\mu\) to \(\mu+\sigma\), and the total within \(\mu\pm\sigma\) is \(68\%\), half of that is \(34\%\).

Step7: Use the symmetry of the normal distribution for part g

Since the normal distribution is symmetric about the mean \(\mu = 61\), \(50\%\) of the data is less than the mean.

Step8: Use the properties of the normal distribution for part h

We know that \(50\%\) is less than the mean (\(61\)). The percentage from \(61\) to \(66\) is \(34\%\). So \(50\%+34\% = 84\%\) is less than \(66\).

Answer:

a. \(61\)
b. \(5\)
c. \(68\)
d. \(95\)
e. \(99.7\)
f. \(34\)
g. \(50\)
h. \(84\)